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Considering the hypothetical distance between consecutive reals.

  1. Nov 29, 2012 #1
    Hi,

    I have been pondering about a hypothetical distance between consecutive real numbers. It seems a bit of a paradox, though I expect it will be shown to be a consistent picture. I'll be using recently-learned terminology which hasn't completely set in mind yet, so please have patience with me :)

    Firstly, let's define a hypothetical function S_r from R->R which in essence is the real analog of the successor function from N->N . I realize that such a function is inherently poorly defined since "succeeding" a real number x by some finite, non-zero number y will give z. But of course, between x and z will be an infinite number of real numbers to fill the interval of y-length. Never-the-less, being a hypothetical question, the terminology should serve for purposes of discussion.

    Now, let's say we have an x and y in the set of reals satisfying |x-y|=0. From this, it is apparent to me that we can deduce that x=y. However, what I don't understand is why we can't also deduce that x=S_r(y), or that y=S_r(x). What I mean is that if we define y to be the real successor to x, and if the distance between x and y, |x-y|, were some non-zero quantity, then y couldn't be the real successor to x since the interval [x,y] would contain an infinite number of reals n that satisfy x<n<y. So the distance between x and its successor y cannot be greater than 0. In other words, it must equal zero, or |x-y|=0. But from this equation, we were also able to deduce that x=y.

    Suppose there were a way to rigorously define the distance between consecutive reals. This distance could not be non-zero, so it would have to be based on some definition of zero itself, let's call it null for reference. Now, given a rigorously defined S_r, if x=S_r(y), then |x-y|=null=0. And if |x-y|=0, then x=y. But since x is also equal to S_r(y), then x=y=S_r(y), or in other words, y=S_r(y)?

    I feel like I'm dressing up something more simple in unnecessary amounts of symbolism. I guess what I'm trying to say is if the distance two consecutive reals can't be non-zero, then it must be zero? And if so, then a real and its real successor must have the same quantity?

    Thanks for reading!
     
  2. jcsd
  3. Nov 29, 2012 #2

    AlephZero

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    The ideas of "consecutive real numbers" and a "successor function for real numbers" don't make sense. For any two real numbers a and b, there is another real number in between them, for eaxmple (a+b)/2.
     
  4. Nov 29, 2012 #3
    I mentioned this as well, the fact that such a distance between consecutive reals would have to be based on some definition of zero itself, provided a rigorous definition. Is there no basis to think of it hypothetically?
     
  5. Nov 29, 2012 #4
    There is no such function, so asking about the properties is pointless (and certainly won't reveal anything about the actual real numbers). Given a real number, there is no other real that could reasonably be identified as its "successor".
     
  6. Nov 29, 2012 #5

    micromass

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    No, there is no basis to think of hypothetically. A notion of "consecutive real numbers" is something that does not exists. So we can't talk about it
    Furthermore, this forum is not the place to talk about hypothetical, non-existent notions.

    Is there something you would like to ask or discuss that is not just merely hypothetical?
     
  7. Nov 29, 2012 #6
    Touché :P But joking aside, I'll respect that this forum probably isn't the best place for this discussion.
     
  8. Nov 29, 2012 #7

    HallsofIvy

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    What "joking" are you referring to? Do you understand what "consecutive real numbers" would have to mean and why there are no "consecutive real numbers"?

    It's not just a matter of this forum not being the ""best place for this discussion". There is NO discussion. What would you think about a discussion of "blue real numbers"?
     
  9. Nov 29, 2012 #8
    This is just a misunderstanding. When micromass said "Is there something you would like to ask or discuss that is not just merely hypothetical?" I took it as a joke thinking he was highlighting the fact that my posts have a tendency of going "off the rails" to the hypothetical. So I said Touché. And when he said "Furthermore, this forum is not the place to talk about hypothetical, non-existent notions", I agreed, and that's all, there's no subtext here.

    And I do understand why there cannot be consecutive real numbers, assuming they exist produces an inconsistent result after all (that the a real and its consecutive real are the same value) as I questioned about in my OP. I was asking this because I'm trying to explore multiple trains of thought related to zero and its arithmetic properties, and this idea seemed relevant.
     
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